More types of Errors - Units & Measurements Video Lecture - Class 11

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part 6 next we will discuss least count
error what is least count these count is
the smallest value an instrument can
measure it is the smallest value at
instrument can measure for example you
must be having a scale at your home the
normal scale which you use in your
geometry box if you observe that there
would be divisions like this I just draw
a rough figure to make you understand
for example you would have something
like this so this is the smallest
division that your scale can measure
there is no subdivisions inside this now
let us suppose you want to measure up
you may want to measure the length of an
object which is lesser than this small
division then you cannot measure that
which you are need to scale correct so
this division this is the least count
that means this is the smallest value
that your either see can't measure so
when this top of least count error we
mean to say that a piece of certain
instruments the small division of the
main scale doesn't coincide with the
small division of the fermier scale the
best example of least count error would
be the vernier calipers what is pioneer
calipers it is an instrument which is
used to measure the thickness or length
of very thin objects which we normally
cannot measure with our meter scale so
in vernier calipers we have a means
this is the main scale and there is a
vernier scale so least count error
arises when the zero of the main scale
and the zero of the vernier scale do not
consent that means how we define these
count error is one means a delusion
minus one vernier scale division that is
the least found error so let us
understand it once again these count
means the smallest value that an
instrument can measure these count error
arises when the smallest value that an
instrument can measure has certain
uncertainty involved in it for example
the vernier calipers it has two scales
the main scale and the vernier scale the
least count of this instrument would be
where the zeros of zero of the vernier
scale would coincide with that of the
main scale so zero of the main scale
would coincide with the zero of the
binary scale if they coincide exactly
then there is no least found error but
if they do not coincide exactly then
there is a least count error involved so
that is the least count error now let us
now discuss certain kinds of ever like
the absolute error absolute error is the
difference between individual
measurement and true value of a quantity
that is the measurement which we take
and the actual value of the quantity
which we are measuring normally it is
denoted by modulus of Delta a since it
is modulus therefore it is always
positive
modulus of any quantity is always
positive let us take an example
suppose we have any quantity which we
have measured 10 times now if I ask you
what is the absolute error involved in
each of the measurements then you would
say this is the absolute error involved
involved in the first measurement which
is the actual measurement minus the true
value of the quantity
similarly the absolute error in the
second measurement is the measured value
minus the true value and so on so that
is absolute error let us now discuss
relative error when I say relative error
it is the ratio of mean absolute error
to mean value of the quantity measured
for example let us suppose we have a
quantity which we discussed in case of
absolute error so we measure that
quantity some 10 times so the absolute
we measured the quantity 10 times now
the absolute errors involved in the
measurement side
similarly in the second measurement the
absolute error would be this similarly
in the third measurement it would come
out to be this and so on me ABS when I
say mean absolute error that means it is
the mean of all these absolute errors so
the mean absolute error would be equal
to Delta K 1 plus Delta a 2 plus dot dot
dot til Delta a 10 this divided by 10 so
this will be that mean absolute error so
relative error is this mean absolute
error so this will be the value of Delta
a in beam so relative error is this
value divided by the mean of the actual
quantity here
okay the next is percentage error
percentage error is nothing but the
relative error expressed in terms of
percentage that is relative error in
terms of percentage first you calculate
relative either this this was your
relative error and then multiply it by
hundred
it is normally denoted by Delta e so
what did we study we studied absolute
error which is denoted by modulus of
Delta a we studied relative error which
is Delta a mean divided by a mean and we
studied percentage error that is
relative error in 200 and this is
denoted by Delta e
now let us look at a few numerical
problems now in successive measurements
of period of oscillation of a pendulum
the readings turn out to be two point
six three seconds two by five six
seconds
two point four two seconds two point
seven one seconds and two point eight
zero seconds calculate the absolute
errors relative error and percentage
error okay so let us now this first
calculate absolute and we know for
absolute error how would you how there
are one two three four five five
measurements so the absolute errors we
will calculate like this Delta a 1 would
be equal to modulus of the other any one
will be equal to a 1 minus a me correct
now what will be the value of a mean let
us calculate the value of a mean a mean
would be two point six three plus two
point five six plus two point four two
plus two point seven one plus two point
eight zero this divided by five this
will come out to be two point six two
seconds now we know ami so Delta a1
would be equal to two point six three
minus two point six three so this will
come out to is 0.01 seconds similarly we
will calculate Delta a which will be a
two minus aiming so that will be equal
to two point five six minus two point
six two this will be equal to zero point
zero
six seconds again Delta a three will be
a 3 minus a mean that is equal to two
point four two minus two point six two
that comes out to be zero point two zero
seconds again Delta a 4 that is a four
minus a which is equal to two point
seven one minus two point six two which
is equal to zero point zero nine seconds
and finally Delta a 5 that is equal to a
5 minus a mean and this is equal to two
point eight zero minus two point six two
which is equal to zero point one eight
seconds so these are the values of
absolute errors so we got the value of
the absolute errors now we have to
calculate the relative error now as we
know relative error is equal to Delta a
mean divided by a mean now we have to
calculate the mean value of all the
absolute errors so let us do that first
so Delta a mean be equal to 0.01 we will
take average of all these values which
we obtain that the 0.01 plus zero point
zero 6 plus zero point two zero plus
zero point zero 9 plus zero point one
eight divided by five so this comes out
to be zero point five one seconds
so this so therefore relative error will
be equal to 0.1 1 divided by a mean so a
minus 2.6 2 so this divided by 2 point 6
2 therefore this comes out to be 0.04 so
this is the value of the relative error
therefore the percentage error would be
relative error multiply it by untoned so
this is the relative error multiply it
by 100 so it comes out to be 4 percent
here so we derived all the values for
absolute error for relative error and
for the percentage error I hope it is
clear to you now that water the three
types of error which we normally need to
calculate and how do we put them into
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